Non-linear analysis of composite beams under positivebending

G. Fabbrocino, G. Manfredi *, E. Cosenza

Dipartimento di Analisi e Progettazione Strutturale, UniversitaÁ Federico II, Napoli, Italy

Received 4 March 1997; received in revised form 4 August 1998

Abstract

The structural behaviour of steel±concrete composite beams depends on the interaction between the steel beam

and the concrete slab. Therefore, the connection largely in¯uences the global behaviour of the beam and itsmodelling is a key issue in the analysis of these structures. An e�ective model requires the introduction of anexplicit relationship between slip and interaction force given by each connector, which is strongly non-linear. In thispaper, a numerical procedure that allows a reliable analysis of the structural behaviour of composite beams

subjected to sagging moment due to short term loads, both in serviceability and ultimate state, is proposed. # 1998Elsevier Science Ltd. All rights reserved.

Keywords: Non-linear analysis; Steel-concrete beams; Structural behavior

1. Introduction

The use of composite steel±concrete structures

results in optimum performance of the two materials;

but in the design process it is required to evaluate the

in¯uence of the connection on the structural behaviour

of the element. In fact, as is suggested by the most

advanced codes [1], the design may be based on the

concept of partial shear connection, and, therefore, the

slip between the concrete slab and the steel beam can-

not be neglected [2]. Furthermore, in the plastic ®eld, a

large amount of the energy dissipation is related to the

connection.

Starting from these remarks, it is evident that:

1. the modelling of the structural behaviour of steel±

concrete beams is complex, and must necessarily

consider the slip between the concrete slab and the

steel beam and the constitutive relationship of the

shear connectors, which is strongly non-linear [3];

2. the behaviour in terms of deformability andstrength of composite beams depends directly on

the distribution of the slip and the consequent inter-action force between the slab and the steel beam;

3. under sustained loads, creep and shrinkage cana�ect the de¯ection, [4, 5].

Therefore, to have a reliable model, these phenomenamust be taken into account [6, 7]. In this paper a nu-

merical procedure, based on well-known kinematicmodels [8, 9] and suitable to analyse the behaviour ofcomposite beams under short term loads, is proposed.The introduction of an original moment±curvature

generalised relationship for the cross section allowseasy analysis of the structural behaviour, varying theconnection level and the material properties in order to

optimise the design in serviceability and ultimate state.

2. The structural modelling

The composite beam consists of two components,

identi®ed in Fig. 1(b) by the subscript s for the lowersteel beam and c for the concrete slab. The two com-ponents are characterised by the same displacement: in

Computers and Structures 70 (1999) 77±89

0045-7949/99/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00173-4

PERGAMON

* Corresponding author. Tel: 0039 81 768 3488; Fax: 0039

81 768 3491; E-mail: gamanfre@unina.it.

this way the two parts have the same rotation and the

same curvature in each section of the beam [8, 9].

In a more re®ned model, each component can be

regarded as a distinct beam, so that the two curvatures

may be di�erent, as the compatibility of the vertical

displacements may be imposed only in the sections

containing the shear connectors [10, 11]. Thus an uplift

of the slab may occur; nevertheless, this e�ect is not

relevant, as Johnson and Molenstra [12] have demon-

strated, and it will be neglected in the following. The

shear connectors are considered in their actual position

along the beam, so, from a static point of view, their

action consists of an interaction force applied at a dis-crete number of sections. The concrete slab is modelled

as a conventional reinforced concrete element, so itsanalysis is based on the two following hypotheses:

1. there is no slip between reinforcing bars and con-

crete;

2. concrete in tension is neglected.

These hypotheses are also acceptable at the ultimate

state; in fact analysis of the section behaviour shows

Fig. 1. The structural model.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±8978

that the concrete slab is mainly in compression and,

whenever it is partially in tension, the steel strain is

small and comparable with that of the concrete.

Therefore, a more re®ned model for the reinforcing

steel in tension, including the slip between rebars and

concrete in tension [13], is not required when the com-

posite section is subjected to sagging moment.

The evaluation of the displacements w(x, y) of the

section not only depends on the rotation j and the dis-

placement of one of the centroids [ws, or wc, Fig. 1(c)],

but also on the slip between the slab and the steel pro-

®le:

s � wups ÿ wlo

c : �1�The ®rst term w up

s is the displacement of the upper

®ber of the steel beam, while w loc is the displacement of

the lower ®ber of the concrete slab.

The slip can be evaluated as a function of the displa-

cements of the centroid of each part:

s � �ws � f � ds� ÿ �wc ÿ f � dc� � ws ÿ wc � f � d �2�where j is the rotation, the same for the two parts of

the section; ds and dc are the distances between the

centroid axes of steel or concrete and the steel concrete

interface, and d is the distance between the two axes.

Eq. (2) can be derived with respect to the position of

the section along the beam, providing an equation con-

taining the curvature, the strain at the centroid of each

part and the derivative of the slip:

ds

dz� dws

dzÿ dwc

dz� w � d � Es ÿ Ec � w � d: �3�

Eqs. (1)±(3) represent the condition of compatibility,

which must be satis®ed at the interface between the

two parts of the cross section.

From a kinematic point of view, it is clear that the

solution depends on three variables: the displacements

of the two centroids and the rotation of the section, or

the strains of the two centroids and the curvature. For

equilibrium of the section, three equations govern:

. global rotational equilibrium of the cross section;

. translational equilibrium of the concrete slab;

. translational equilibrium of the steel pro®le.

Eq. (4) represents the rotational equilibrium of the sec-

tion evaluated with respect to the ®ber of contact

between the two parts of the beam:

M �Ms �Mc � Fs � ds ÿ Fc � dc: �4�

In this equation M is the global bending moment

acting on the section, Ms and Mc are the two parts of

the global bending moment acting on the steel beam,

and the concrete slab; Fs and Fc are the axial forces

applied on the steel beam and the concrete slab, re-

spectively. It is assumed that both axial force and

bending moment are applied at the centroid of each

component of the cross section.

The partial translational equilibrium of the concrete

slab gives:

Fc � ÿF: �5�

The partial translational equilibrium of the steel

beam gives:

Fs � F: �6�

From global translational equilibrium, Eq. (4) can

be modi®ed into the more useful form

Fs � ÿFc � F �7�so that global rotational equilibrium can be rewritten:

M �Ms �Mc � F � d: �8�

In this way the three independent Eqs. (5), (6), (8)

depend on the interaction force F, and the two bending

moments acting on the two parts of the composite sec-

tion.

These forces are the resultants of the normal stresses

acting on the two part of the section:

Fs ��As

ss�x; y� � dA; �9a�

Ms ��As

ss�x; y� � �yÿ ds� dA �9b�

Fc ��Ac

sc�x; y� � dAÿXj

Ajrs � sjrs; �9c�

Mc ��Ac

sc�x; y� � �yÿ dc� � dA

�Xj

sjrs � Ajrs � �yjrs ÿ dc�; �9d�

F �Xn�x�j�1

Fj: �9e�

The subscript rs de®nes the reinforcing steel; y rep-

resents the distance between each ®ber and the cen-

troid of the global section [see Fig. 1(c)]; Fj is the force

acting on connector j placed before the current section,

n(x) is the number of shear connectors, jR n(x).

Statically, the e�ect of the connection is represented

by the term F � d, therefore even if the bending

moment acting on each part of the section is equal to

zero (zero curvature), the composite section is able to

bear a global bending moment, depending on the value

of the interaction force F.

This remark is very signi®cant as the bearing ca-

pacity and thus the moment±curvature relationship

depends on the value of the interaction force.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±89 79

3. The constitutive relationships

The above equations are not su�cient and the con-

stitutive relationships of the materials must be intro-

duced for the concrete in compression, for steel

reinforcement, the structural steel, and the shear con-

nectors. It is worth noting that the numerical pro-

cedure developed is not strictly related to the

constitutive laws discussed below and used to perform

the theoretical±experimental comparison, as even ex-

perimental stress±strain relationships can be used.

The mechanical behaviour of the reinforcing steel of

the slab is modelled using the Ramberg±Osgood for-

mulation, which is suitable for simulating the real

mechanical behaviour of cold-formed, tempered, or

low content of carbon steel. For concrete in com-

pression, the relationship given by Mander et al. [14]

has been used; the descending branch has been modi-

®ed as suggested by Hilleborg [15] in order to take

into account the size e�ect.

The proposed method regards only beams subjected

to sagging moment, so the buckling phenomena are

not signi®cant. A simple three-linear idealization, as

shown in Fig. 2 models the structural steel. In the

same ®gure the main parameters, needed to de®ne the

relationship, are shown.

For the shear connection consisting of headed studs,

the relationship de®ned by Olgaard [3] is used; it is

characterised by strong non-linearity and has been

extensively used. It is described by two independent

parameters: a and b. Some curves representing the

mechanical behaviour of the headed studs are plotted

in Fig. 3. They have been de®ned taking into account

typical values of the two coe�cients found in

literature [9, 12]: a=0.558 and b=1 mmÿ1 (type A),

a=0.989 and b=1.535 mmÿ1 (type B), or a=0.8and b=0.7 mmÿ1 (type C).

The relation between force Fj and slip sj assumes thefollowing form:

Fj � Pmax � �1ÿ eÿ b � sj�a: �12�

The two parameters a and b control the initial slopeof the curve and its shape, while the value of Pmax has

to be measured by means of push-out tests [1].

4. Moment curvature relationship of the cross section

The global bending moment on the composite sec-tion is given by axial force and bending acting on each

part of the section. Therefore, for a particular value ofthe curvature [Eqs. (3), (8), (9)] the value of the globalbending moment depends on the derivative of the slips, Eq. (3). Similarly, the curvature depends directly on

the value of the interaction force F, Eq. (8), if the glo-bal bending moment is given.It is therefore not possible to draw a single

moment±curvature curve, conversely, it is necessary tode®ne a family of curves corresponding to di�erentvalues of F. Thus, the complete solution of the ¯exural

problem of a composite section can be obtained arran-ging an array with this family of curves.In order to de®ne this family of curves, ®rstly, the

de®nition of the upper and lower limit of the inter-

action force F, positive if it gives compression on theslab, is required. The upper positive limit of the inter-action force Fmax is given by the minimum value

between the ultimate compressive axial force of theslab and the ultimate tension axial force of the steelpro®le. The lower negative limit Fmin is given by the

Fig. 2. Constitutive relationship for structural steel.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±8980

maximum value between the ultimate tension axialforce of the slab and the ultimate compressive axial

force of the steel pro®le. These boundary values are

easy to be calculated and depend on the mechanical

properties of the coupling of the concrete slab and the

steel beam.

In the following, some numerical results are shown,

with reference to a tested beam called PI4 [11]; the

geometrical and mechanical properties of the cross sec-

tion are given in Fig. 4; and the constitutive laws used

are those previously described.

The complete solution of the ¯exural problem of thecomposite section is given in the graph of Fig. 6,

where two di�erent families of curves are plotted. The

continuous lines represent the moment curvature re-

lation for a particular value of the interaction force;

the dotted ones represent the moment±curvature

curves with a given value of the slip derivative ds/dz.

The last family of curves is very interesting, in fact

the curve corresponding to the value of the derivative

ds/dz equal to zero de®nes the moment±curvature re-

lation for the full interaction condition. The curve is

drawn using a thick line in the same ®gure and a circle

marks the corresponding ultimate state condition. This

curve is the only one consistent with the Bernoulli hy-

pothesis of linear deformation of the entire composite

cross section, which occurs when the slip and the de-

rivative at the slab pro®le interface are both zero. In

turn, the grey thick line plotted in the same ®gure

gives the moment±curvature relationship for the com-

posite section in absence of interaction (F=0); in this

particular condition, the curvature is related to the

bending moment by the following relation:

w � ÿ M

�EsIs � EcIc� �13�

where EsIs and EcIc are the ¯exural rigidities of the

steel pro®le and of the concrete slab [8].

The analysis of the curves plotted in Fig. 5 allows

some interesting remarks:

1. The ultimate curvature for the full interaction con-

dition is considerably lower than the maximum

values allowable when the two parts of the beam

can slip; the large in¯uence of the connection beha-

viour on the plastic deformation of the beam is

clear, and can be obtained by an appropriate design

of the strength and deformability of the shear con-

nectors.

2. The section subjected to positive bending may be

characterised by both negative and positive curva-

ture, as a function of the interaction force level. For

example it can be observed that when the bending

moment is low enough and, at the same time, the

concentrated action of the connectors provide a

large interaction force, tensile stresses can arise in

the upper ®bers of the slab, even though the cross-

section is under positive bending. This situation

usually occurs in the support regions of the simple

supported beams.

Fig. 3. Constitutive relationship for headed stud

connectors [3].

Fig. 4. Cross-section and material properties of beam PI4 [11].

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±89 81

3. When the value of the curvature is zero, the sectionis able to bear a non-zero bending moment, depend-

ing on the interaction level. In fact, as shown in

Eq. (8), when the curvature is zero the two bending

moments acting on each part of the section areequal to zero, but the term F�d is not zero, as it is a

linear function of the interaction force F.

4. The maximum positive bending moment which may

be borne by the section is given by the relation

Mmax=Fmax�d, where Fmax is the maximum positive

interaction force consistent with the mechanicalproperties of each part of the section. Therefore,

only a curvature equal to zero may be associated

with this value of the bending stress. This remark is

clearly shown in Fig. 5, where the greater the inter-action force, the lower the curvature range, so that

the moment±curvature curves tend to become a

point, corresponding to the co-ordinates w=0,M= Mmax=736.94 kN m, and marked by a tri-angle in the graph.

5. as F increases, a change in the shape of the M±wcurves can be observed; beyond a certain value ofthe interaction force, the moment±curvature re-lation does not cross the curvature axis and a non

zero lower limit of the bending moment exists. Thisbehaviour is better shown in Fig. 6, where typicalmoment±curvature relationships are plotted; the

®rst one, named type 1 curve, crosses the curvatureaxis, the second one, type 2 curve, shows a non zerovalue of the minimum bending moment Mmin(F).

The moment±curvature generalised relationship is avery useful tool in the design process, as it is possibleto de®ne the curvature of the section and the corre-

sponding value of the derivative ds/dz if both the valueof the interaction force F and the global bendingmoment are given.

5. The analysis of the beam

The solution of the equations governing the problem

is quite di�cult. In fact, the constitutive relationshipsare strongly non linear and also particular conditionsof coupling in the equations are present. Therefore, the

solution can be obtained only using a numerical pro-cedure. The solution of the entire beam at each stepmust be performed, with a signi®cant computational

e�ort.In general the boundary conditions can be written

corresponding to the extreme sections of the beamFig. 6. Typical moment±curvature curves.

Fig. 5. Moment-curvature generalised relationship.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±8982

where two situations may occur: the connector can be

present or not. In the ®rst case, the interaction force is

equal to zero, in the second one the slip and the inter-

action force must satisfy the Ollgaard's constitutive re-

lationship (12).

In order to simplify the discussion, a case of a beam

characterised by both geometrical and mechanical sym-

metry will be considered. Moreover, changing only the

boundary conditions can solve any di�erent problem.

Due to the symmetry, the analysis can be restricted to

only a half beam. The boundary conditions can be

given in the ®rst section of the beam, where the slip

and the interaction force are related as discussed

before and in the section placed on the symmetry axis,

where the slip between the slab and the steel beam

must be equal to zero.

A reliable numerical solution of the equations may

be obtained using the ®nite di�erence method [16],

combined with a shooting type procedure, which allows

the transformation of a boundary conditions problem

into an initial value one. This approach allows trans-

forming the resolution of the problem into the search

of the root of a non-linear equation, which can be

evaluated by means of an iterative procedure. The

independent variable of this equation is the value of

the slip s in the ®rst pivotal section. In the numerical

procedure the half beam can be divided into n parts,

de®ned by a constant spacing of the nodes equal to

Dz. The only unknown parameter is the value assumed

by the function s(z) in each integration node.

In particular, the two following equations can be

used:

si�1j � sij �ds

dz

� �i

j

�Dz; �14a�

F i�1j � F i

j � �F ij �14b�

where j gives the iteration number, and i the node

number. Eq. (14a) gives the slip, while Eq. (14b) gives

the interaction force present in the node i+1. The

parameter F ij depends on the presence of a shear con-

nector at the node i. In fact, when there is not any

shear connector the parameter F ij is equal to zero, con-

versely if a shear connector is present, F ij has to be

evaluated by means of the non linear constitutive re-

lationship (12).

The value assumed in the node i by the derivative

ds/dz can be simply calculated. In fact, as the global

bending moment and the interaction force are known

in the current section, the corresponding value ds/dz

can be evaluated by a simple linear interpolation of

the data available in the moment±curvature array.

Therefore, the procedure allows de®nition of the

unknown value s i+1j in each node and so also in the

last node n+1, which is assumed as the convergenceparameter of the iterative process.

6. Computational aspects of the procedure

The condition governing the convergence may beexpressed by the equation:

E�s1j � � sn�1�s1j � � 0: �15�

An upper and a lower limit of the root of Eq. (15)may be de®ned, in order to restrict the range of thetentative values. In fact, the lower limit is given by the

full interaction condition. In this situation there is noslip between slab and steel beam:

s1min � 0: �16�

The upper limit is reached when shear connectorsare not provided at the slab±pro®le interface, there-

fore, it is given by the product of the rotation of the®rst section in absence of interaction and the distanced, see Fig. 1(c).

The above upper limit of the slip requires the evalu-ation of the rotation at the support; another e�ectiveupper limit for the slip in the ®rst integration node canbe evaluated in an easier way, extracting the maximum

modulus of the derivative ds/dz from the generalisedmoment±curvature relationship.Therefore, it can be assumed that the maximum

value of the slip corresponding to the ®rst integrationnode is given by:

s1max �L

2����� dsdz

����max

: �17�

This value is calculated assuming that the derivativeds/dz is constant; this is certainly the absolute maxi-

mum of the slip present in the ®rst section of thebeam.Typical shapes of the function E(s1) have been

plotted in Fig. 7. Four di�erent curves have beende®ned for a particular load. They depend on the levelof interaction, de®ned by the ratio N/Nf , where N is

the number of the shear connectors present along thebeam and Nf is the number strictly needed to ensurethe full interaction. The spacing of shear connectors isconstant along the beam. The curves represent the

function E(s1) throughout the entire existence domainS. It is clearly characterised by an upper and a lowerlimit depending on the ratio N/Nf , which are not co-

incident with the two limits discussed before.Nevertheless, the existence domain always strictlysatis®es the following condition:

S 2 �0; s1max�: �18�

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±89 83

This remark is obviously essential to schedule a re-

liable solution scheme.

It is also worth noting that the greater the ratio

N/Nf , the steeper the error function; therefore, the nu-

merical solution is more di�cult to obtain, when the

beam tends to the condition of full interaction.

Besides taking into account Fig. 7, it can be

observed that:

1. The upper limit of the tentative slip, marked by a

triangle, represents a limit state for an intermediate

section i, a�ected by an interaction force F corre-

sponding to a type 2 moment±curvature relation

(see Fig. 6) and a bending moment Mi lower than

the minimum consistent with the interaction level,

Mmin(F). Therefore, the evaluation of the curvature

and the derivative ds/dz is not possible because the

moment±curvature curve does not cross the hori-

zontal line representing the value of the bending

moment: whenever this situation occurs, the conver-

gence parameter cannot be calculated.

2. The lower limit of the tentative slip, marked by a

circle, is due to two possible limit states, depending

on the load level. The ®rst one is reached when a

negative value of the interaction force exceeds in

modulus the limit vFminv, de®ned in the last section;

the second one is reached when the global bending

moment exceeds the maximum value consistent with

the interaction force Mmax(F), Fig. 6.

The uniqueness of the solution in this structural pro-

blem ensures that the above critical conditions cannot

be replaced or modi®ed. So the values of the tentative

slip not included in the interval are not consistent with

the model and consequently are not possible.

These conditions must be considered to solve the

problem by means of a processing program, as the

existence domain of the function E(s1) cannot be pre-

viously de®ned. The numerical process assumes that a

discrete number of (w, M, ds/dz) curves must be calcu-

lated, so that an array is available for each interaction

level considered.

Thus, for a given bending moment and an inter-

action force, the corresponding values of the curvature

and of the derivative ds/dz can be evaluated by means

of a simple linear interpolation. Moreover, this

approach allows the minimisation of the time of calcu-

lation, in fact, during the iterations, the solution of the

complex problem of translational and rotational equili-

brium of the section in each integration node is not

requested.

The beams are statically determined, thus the global

bending moment is known in each node; therefore, the

fundamental steps of the procedure may be summar-

ised, in compliance with the ¯ow-chart drawn in Fig. 8:

1. de®nition of a suitable moment±curvature relation-

ship for the cross section of the composite beam to

be analysed;

2. evaluation of the maximum value of the tentative

slip between the slab and the steel pro®le, according

to Eqs. (16) and (17);

3. selection of a ®rst tentative slip to start the pro-

cedure;

4. calculation of the corresponding interaction force in

the section by means of the constitutive relationship

of the headed studs;

5. de®nition of the (M, w, ds/dz) relationship corre-

sponding to the calculated interaction force by

means of a linear interpolation;

6. evaluation of the curvature and the derivative of the

slip related to the given value of the bending

moment;

7. evaluation of the slip in the next node by Eq. (14a).

The knowledge of an interval containing the root of

Eq. (15) allows the use of a bracketing method for the

iterative solution of the problem. Moreover, the iter-

ations of the numerical procedure must consider the

adjustments to the tentative slip related to the critical

conditions de®ned before, whenever the convergence

parameter cannot be evaluated (steps a and b). In fact,

the slip must be reduced when the bending moment of

the section is lower than the value Mmin(F); on the

other hand, the attempt slip has to be increased when

the interaction force becomes inconsistent with its

lower limit, or the bending moment is greater than the

maximum Mmax(F) consistent with the current inter-

action level.

When the convergence condition is satis®ed, the

knowledge of the curvature allows the evaluation, by a

simple integration of the rotation and the vertical dis-

placement corresponding to each integration node

(step c).

Fig. 7. Typical curves of the convergence parameter.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±8984

Fig. 8. Flow-chart of the numerical procedure.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±89 85

7. A theoretical±experimental comparison

The proposed procedure has been validated using

di�erent tests with good results. In the present section

the simulation of a test by Aribert and Aziz [10, 11] is

presented. The beam, called PI4 in Ref. [11], is simply

supported and characterised by the cross-section and

material properties summarised in Fig. 4 and also by

the following geometrical and mechanical properties:

. e�ective span length 5000 mm;

. 18 headed stud connectors Nelson F 19;

. distance between studs 650 mm;

Fig. 9. Numerical and experimental results for the functions: s(z) (a); ds/dz (b); w (c).

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±8986

. stud constitutive relationship parameters:

Pmax=130 kN; a=0.8 e b=0.7 mmÿ1.

The numerical procedure presented before allows the

comparison of both local, such as slip at the slab±steel

beam interface, and global results, such as the load±

de¯ection relation. A complete solution of the struc-

tural scheme has been performed using an integration

interval equal to 5 mm; all the most signi®cant results

have been arranged in Figs. 9±11.

Fig. 9 shows the structural scheme and the plot of

the calculated functions: sÐFig. 9(a); ds/dzÐFig. 9(b);wÐFig. 9(c), for two di�erent load levels. The com-

parison in terms of slip shows good agreement of theexperimental and theoretical results and the following

remarks can be made:

1. the midspan section is never characterised by the

full interaction condition (ds/dz=0), in turn, thegreater the load, the greater the value of the deriva-

tive itself;

Fig. 11. Load±de¯ection curve, theoretical±experimental comparison.

Fig. 10. Calculated moment±curvature relation for the midspan section of beam PI4.

G. Fabbrocino et al. / Computers and Structures 70 (1999) 77±89 87

2. along the beam, it is possible to identify only threesections characterised by the full interaction

(ds/dz=0), Fig. 9(b).

The curvatures are plotted in Fig. 9(c); a zone a�ected

by a negative curvature in the support regions is pre-sent. The procedure allows solving of the problemboth in serviceability and at ultimate state, as it is easyto recognise the yielding of the midspan section and its

subsequent spreading.In this way, the entire load process can be ®tted, the

moment±curvature relationship for each section of the

beam can be obtained, and, in particular for the mid-span section, Fig. 10. The last point of the curve givesthe ultimate state condition for the beam, as it is

placed on the borderline of the moment±curvaturedomain; it is also easy to observe that:

1. the value of the derivative ds/dz is very close to

0.03;2. the value of the curvature is considerably greater

than maximum curvature consistent with the full in-

teraction condition (ds/dz=0).

At any load level, the mechanical behaviour of themidspan section is largely di�erent from the full inter-

action condition.The load±de¯ection curve is plotted in Fig. 11,

where the proposed model is compared with the exper-

imental results. The shapes of the theoretical and ex-perimental curves show good agreement.The experimental collapse has been reached for a

load (Pu)exp=490 kN and a de¯ection equal to

157 mm [11]. The calculated ultimate load is very closeto the experimental one: (Pu)cal=488 kN. On theother hand, for the ultimate de¯ection, the error is

greater, as the calculated ultimate de¯ection is 182 mm,with a scatter equal about to 16%. Therefore, theevaluation of the ultimate de¯ection is very di�cult, as

it depends strongly on the constitutive relationships ofboth the materials and the shear connectors.

8. Conclusions

The modelling of the structural behaviour of steel±concrete composite beams under short term loads has

been discussed in this paper. In particular, it has beenpresented an original approach, based on the extensionof the concept of moment±curvature relationship to

the composite cross-sections.The numerical procedure shows a good convergence

and some indications are provided on the choice of the

tentative values for the slip in the iterative solution.The proposed method allows a reliable analysis of thesteel±concrete composite beams, demonstrated by a

wide theoretical±experimental comparison. The theor-etical results show a substantial agreement with the ex-

perimental data in terms of both local and globalquantities. Nevertheless, it can be remarked that, evenif the results in terms of ultimate bearing capacity are

very good, the estimation of the ultimate de¯ectionsseems to be more in¯uenced by the constitutive re-lationships introduced for the materials and the shear

connectors.The structural behaviour is quite complex, but the

proposed model is suitable for describing the main

mechanical phenomena. Furthermore, the proposedmoment±curvature generalised relationship is a verye�ective tool for understanding the structural beha-viour and also to perform an optimisation of the de-

sign in terms of both strength and deformability forthe steel-concrete composite beams.

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